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Properties

Label 675.5.c.h.26.2
Level $675$
Weight $5$
Character 675.26
Analytic conductor $69.775$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,5,Mod(26,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.26"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 675.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,14,0,0,38] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(69.7747250816\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 26.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 675.26
Dual form 675.5.c.h.26.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{2} +7.00000 q^{4} +19.0000 q^{7} +69.0000i q^{8} +123.000i q^{11} -302.000 q^{13} +57.0000i q^{14} -95.0000 q^{16} -414.000i q^{17} -304.000 q^{19} -369.000 q^{22} -300.000i q^{23} -906.000i q^{26} +133.000 q^{28} -678.000i q^{29} +239.000 q^{31} +819.000i q^{32} +1242.00 q^{34} -740.000 q^{37} -912.000i q^{38} +228.000i q^{41} +982.000 q^{43} +861.000i q^{44} +900.000 q^{46} -2166.00i q^{47} -2040.00 q^{49} -2114.00 q^{52} +1593.00i q^{53} +1311.00i q^{56} +2034.00 q^{58} -2922.00i q^{59} -316.000 q^{61} +717.000i q^{62} -3977.00 q^{64} -4622.00 q^{67} -2898.00i q^{68} +1818.00i q^{71} +3031.00 q^{73} -2220.00i q^{74} -2128.00 q^{76} +2337.00i q^{77} -10450.0 q^{79} -684.000 q^{82} -12633.0i q^{83} +2946.00i q^{86} -8487.00 q^{88} -7002.00i q^{89} -5738.00 q^{91} -2100.00i q^{92} +6498.00 q^{94} +6517.00 q^{97} -6120.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + 38 q^{7} - 604 q^{13} - 190 q^{16} - 608 q^{19} - 738 q^{22} + 266 q^{28} + 478 q^{31} + 2484 q^{34} - 1480 q^{37} + 1964 q^{43} + 1800 q^{46} - 4080 q^{49} - 4228 q^{52} + 4068 q^{58}+ \cdots + 13034 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000i 0.750000i 0.927025 + 0.375000i \(0.122357\pi\)
−0.927025 + 0.375000i \(0.877643\pi\)
\(3\) 0 0
\(4\) 7.00000 0.437500
\(5\) 0 0
\(6\) 0 0
\(7\) 19.0000 0.387755 0.193878 − 0.981026i \(-0.437894\pi\)
0.193878 + 0.981026i \(0.437894\pi\)
\(8\) 69.0000i 1.07812i
\(9\) 0 0
\(10\) 0 0
\(11\) 123.000i 1.01653i 0.861201 + 0.508264i \(0.169713\pi\)
−0.861201 + 0.508264i \(0.830287\pi\)
\(12\) 0 0
\(13\) −302.000 −1.78698 −0.893491 − 0.449081i \(-0.851752\pi\)
−0.893491 + 0.449081i \(0.851752\pi\)
\(14\) 57.0000i 0.290816i
\(15\) 0 0
\(16\) −95.0000 −0.371094
\(17\) − 414.000i − 1.43253i −0.697830 − 0.716263i \(-0.745851\pi\)
0.697830 − 0.716263i \(-0.254149\pi\)
\(18\) 0 0
\(19\) −304.000 −0.842105 −0.421053 − 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −369.000 −0.762397
\(23\) − 300.000i − 0.567108i −0.958956 − 0.283554i \(-0.908487\pi\)
0.958956 − 0.283554i \(-0.0915135\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 906.000i − 1.34024i
\(27\) 0 0
\(28\) 133.000 0.169643
\(29\) − 678.000i − 0.806183i −0.915160 − 0.403092i \(-0.867936\pi\)
0.915160 − 0.403092i \(-0.132064\pi\)
\(30\) 0 0
\(31\) 239.000 0.248699 0.124350 − 0.992238i \(-0.460316\pi\)
0.124350 + 0.992238i \(0.460316\pi\)
\(32\) 819.000i 0.799805i
\(33\) 0 0
\(34\) 1242.00 1.07439
\(35\) 0 0
\(36\) 0 0
\(37\) −740.000 −0.540541 −0.270270 − 0.962784i \(-0.587113\pi\)
−0.270270 + 0.962784i \(0.587113\pi\)
\(38\) − 912.000i − 0.631579i
\(39\) 0 0
\(40\) 0 0
\(41\) 228.000i 0.135634i 0.997698 + 0.0678168i \(0.0216033\pi\)
−0.997698 + 0.0678168i \(0.978397\pi\)
\(42\) 0 0
\(43\) 982.000 0.531098 0.265549 − 0.964097i \(-0.414447\pi\)
0.265549 + 0.964097i \(0.414447\pi\)
\(44\) 861.000i 0.444731i
\(45\) 0 0
\(46\) 900.000 0.425331
\(47\) − 2166.00i − 0.980534i −0.871572 − 0.490267i \(-0.836899\pi\)
0.871572 − 0.490267i \(-0.163101\pi\)
\(48\) 0 0
\(49\) −2040.00 −0.849646
\(50\) 0 0
\(51\) 0 0
\(52\) −2114.00 −0.781805
\(53\) 1593.00i 0.567106i 0.958957 + 0.283553i \(0.0915131\pi\)
−0.958957 + 0.283553i \(0.908487\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1311.00i 0.418048i
\(57\) 0 0
\(58\) 2034.00 0.604637
\(59\) − 2922.00i − 0.839414i −0.907660 − 0.419707i \(-0.862133\pi\)
0.907660 − 0.419707i \(-0.137867\pi\)
\(60\) 0 0
\(61\) −316.000 −0.0849234 −0.0424617 − 0.999098i \(-0.513520\pi\)
−0.0424617 + 0.999098i \(0.513520\pi\)
\(62\) 717.000i 0.186524i
\(63\) 0 0
\(64\) −3977.00 −0.970947
\(65\) 0 0
\(66\) 0 0
\(67\) −4622.00 −1.02963 −0.514814 − 0.857302i \(-0.672139\pi\)
−0.514814 + 0.857302i \(0.672139\pi\)
\(68\) − 2898.00i − 0.626730i
\(69\) 0 0
\(70\) 0 0
\(71\) 1818.00i 0.360643i 0.983608 + 0.180321i \(0.0577138\pi\)
−0.983608 + 0.180321i \(0.942286\pi\)
\(72\) 0 0
\(73\) 3031.00 0.568775 0.284387 − 0.958709i \(-0.408210\pi\)
0.284387 + 0.958709i \(0.408210\pi\)
\(74\) − 2220.00i − 0.405405i
\(75\) 0 0
\(76\) −2128.00 −0.368421
\(77\) 2337.00i 0.394164i
\(78\) 0 0
\(79\) −10450.0 −1.67441 −0.837206 − 0.546888i \(-0.815812\pi\)
−0.837206 + 0.546888i \(0.815812\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −684.000 −0.101725
\(83\) − 12633.0i − 1.83379i −0.399125 − 0.916897i \(-0.630686\pi\)
0.399125 − 0.916897i \(-0.369314\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2946.00i 0.398323i
\(87\) 0 0
\(88\) −8487.00 −1.09595
\(89\) − 7002.00i − 0.883979i −0.897020 − 0.441990i \(-0.854273\pi\)
0.897020 − 0.441990i \(-0.145727\pi\)
\(90\) 0 0
\(91\) −5738.00 −0.692911
\(92\) − 2100.00i − 0.248110i
\(93\) 0 0
\(94\) 6498.00 0.735401
\(95\) 0 0
\(96\) 0 0
\(97\) 6517.00 0.692635 0.346317 − 0.938117i \(-0.387432\pi\)
0.346317 + 0.938117i \(0.387432\pi\)
\(98\) − 6120.00i − 0.637234i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 675.5.c.h.26.2 2
3.2 odd 2 inner 675.5.c.h.26.1 2
5.2 odd 4 675.5.d.a.674.2 2
5.3 odd 4 675.5.d.d.674.1 2
5.4 even 2 27.5.b.c.26.1 ✓ 2
15.2 even 4 675.5.d.d.674.2 2
15.8 even 4 675.5.d.a.674.1 2
15.14 odd 2 27.5.b.c.26.2 yes 2
20.19 odd 2 432.5.e.e.161.1 2
45.4 even 6 81.5.d.b.26.2 4
45.14 odd 6 81.5.d.b.26.1 4
45.29 odd 6 81.5.d.b.53.2 4
45.34 even 6 81.5.d.b.53.1 4
60.59 even 2 432.5.e.e.161.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.c.26.1 ✓ 2 5.4 even 2
27.5.b.c.26.2 yes 2 15.14 odd 2
81.5.d.b.26.1 4 45.14 odd 6
81.5.d.b.26.2 4 45.4 even 6
81.5.d.b.53.1 4 45.34 even 6
81.5.d.b.53.2 4 45.29 odd 6
432.5.e.e.161.1 2 20.19 odd 2
432.5.e.e.161.2 2 60.59 even 2
675.5.c.h.26.1 2 3.2 odd 2 inner
675.5.c.h.26.2 2 1.1 even 1 trivial
675.5.d.a.674.1 2 15.8 even 4
675.5.d.a.674.2 2 5.2 odd 4
675.5.d.d.674.1 2 5.3 odd 4
675.5.d.d.674.2 2 15.2 even 4